How many digits are needed?

TL;DR

This paper investigates how many digits in a base-$q$ expansion are needed to approximate a random variable on [0,1), showing convergence of the remainder distribution to uniform and providing bounds on the number of digits required.

Contribution

It establishes convergence of the scaled remainder distribution to uniform and introduces a coupling method for analyzing digit-based approximations.

Findings

Remainder distribution converges to uniform under certain conditions

Coupling method links the remainder to a non-negative integer variable

Exponential convergence rate of the distribution and its PDF

Abstract

Let $X_1,X_2,...$ be the digits in the base-$q$ expansion of a random variable $X$ defined on $[0,1)$ where $q\ge2$ is an integer. For $n=1,2,...$, we study the probability distribution $P_n$ of the (scaled) remainder $T^n(X)=\sum_{k=n+1}^\infty X_k q^{n-k}$: If $X$ has an absolutely continuous CDF then $P_n$ converges in the total variation metric to the Lebesgue measure $\mu$ on the unit interval. Under weak smoothness conditions we establish first a coupling between $X$ and a non-negative integer valued random variable $N$ so that $T^N(X)$ follows $\mu$ and is independent of $(X_1,...,X_N)$, and second exponentially fast convergence of $P_n$ and its PDF $f_n$. We discuss how many digits are needed and show examples of our results. The convergence results are extended to the case of a multivariate random variable defined on a unit cube.